Cindy gets utility from consumption, C, and leisure, L. Her utility function is $U=C+LC$. The price of consumption is $£1$ per unit and the hourly wage rate is $w$. Cindy has no non-labour income. She has to allocate her 24 hour day between leisure and market work, H.
a. What is the relationship between the number of hours she works and the hourly wage rate?
b. How many hours does she choose to work?
I am not sure about part a. I know that generally the higher the wage the more expensive lesiure becomes and assuming that it is a normal good we consume less leisure. But I guess that the question asks for mathematical relation, which I don't know how to show.
I have started by standard maximising utility model:
$U_C$ = $1+L$ and $U_L$ = C then ($C/1+L$) = $w$
then we will have $wH/(1 + (24 - H)$ = $w$
from this I can get $H$ = 12.5 but still didn't get the relationship between $w$ and $H$